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Appendix A: The Stiffness Matrix
Fig. A.2 Sign convention
Fig. A.3 Unit rotation at joint B
developed at B due to unit rotation at B is termed as K 11 , and the moment induced
at C due to this displacement is K 21 Note that there is a relation of the labels 1 and
2 in K 11 and K 21 to joint displacements δ 1 and δ 2 .
In order to determine K 11 , the contribution from both the members AB and BC is
to be taken into account. Referring to Fig. A.3,
K 11 =
4E I
L
+
4E I
L
=
8E I
L
and
K 21 =
2E I
L
Similarly, K 12 =
2E I
L
and K 22 =
4E I
L
.
The stiffness matrix for this problem is
K 11 K 12
K 21 K 22
=
2E I
L
4 1
1 2
The stiffness coefficient K i j is defined as the force developed at point i corresponding to an unknown displacement at i, due to a unit value of the displacement
at j corresponding to an unknown displacement at j.